Theorem pssdifn0 3944

Description: A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)

Assertion

Ref	Expression
pssdifn0	|- ( ( A C_ B /\ A =/= B ) -> ( B \ A ) =/= (/) )

Proof of Theorem pssdifn0

Step	Hyp	Ref	Expression
1		ssdif0 3942	. . . 4 |- ( B C_ A <-> ( B \ A ) = (/) )
2		eqss 3618	. . . . 5 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
3	2	simplbi2 655	. . . 4 |- ( A C_ B -> ( B C_ A -> A = B ) )
4	1, 3	syl5bir 233	. . 3 |- ( A C_ B -> ( ( B \ A ) = (/) -> A = B ) )
5	4	necon3d 2815	. 2 |- ( A C_ B -> ( A =/= B -> ( B \ A ) =/= (/) ) )
6	5	imp 445	1 |- ( ( A C_ B /\ A =/= B ) -> ( B \ A ) =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   \ cdif 3571   C_ wss 3574   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  pssdif  3945  tz7.7  5749  domdifsn  8043  inf3lem3  8527  isf32lem6  9180  fclscf  21829  flimfnfcls  21832  lebnumlem1  22760  lebnumlem2  22761  lebnumlem3  22762  ig1peu  23931  ig1pdvds  23936  divrngidl  33827